Path-Dependent Partial Differential Equations and Optimal Control

路径相关的偏微分方程和最优控制

• 批准号: 2106077
• 负责人: Christian Keller
• 金额: $ 13.3万
• 依托单位: The University of Central Florida Board of Trustees
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-09-01 至 2024-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2106077&HistoricalAwards=false
• 关键词: Path; Dependent; Partial; Differential; Equations

This research will advance optimal control theory to understand realistic and complex scenarios in applications. The research will potentially provide better or even optimal decision-making procedures, especially relevant for problems that affect many stakeholders such as pension fund investments and financial risk management. The project will provide training opportunities for graduate students and opportunities for STEM students to build their careers in the financial industry. The project consists of three parts. In the first part, the investigator will study optimal control problems involving non-Markovian piecewise deterministic processes and the associated non-local Hamilton-Jacobi-Bellman equations. Applications are optimal execution or liquidation problems in mathematical finance with richer classes of models. For example, non-Markovian Hawkes processes can be incorporated. Those processes provide a more accurate description of the relevant financial data. The second part deals with path-dependent Hamilton-Jacobi equations whose Hamiltonians can have quadratic or even super-quadratic growth in the gradient. Those equations are important for optimal control problems with unbounded controls. The investigator will develop new notions of non-smooth solutions and establish well-posedness results with respect to those notions. In the third part of this project, the investigator will completely carry out a path-dependent dynamic programming approach for the optimal control of locally monotone evolution equations. This large class of equations covers the two-dimensional Navier-Stokes equations and the tamed three-dimensional Navier-Stokes equations. In addition, this research provides a new line of methodologies for the largely open problem of optimal synthesis for infinite-dimensional control problems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

这项研究将推进最优控制理论，以理解应用中的现实和复杂场景。该研究将有可能提供更好甚至最优的决策程序，特别是与影响许多利益相关者的问题相关，例如养老基金投资和金融风险管理。该项目将为研究生提供培训机会，并为 STEM 学生提供在金融行业发展职业生涯的机会。该项目由三部分组成。在第一部分中，研究者将研究涉及非马尔可夫分段确定性过程和相关非局部 Hamilton-Jacobi-Bellman 方程的最优控制问题。应用是具有更丰富模型类别的数学金融中的最优执行或清算问题。例如，可以合并非马尔可夫霍克斯过程。这些流程提供了相关财务数据的更准确的描述。第二部分涉及路径相关的 Hamilton-Jacobi 方程，其哈密顿量可以在梯度中具有二次甚至超二次增长。这些方程对于无界控制的最优控制问题非常重要。研究者将开发非光滑解决方案的新概念，并针对这些概念建立适定性结果。在该项目的第三部分中，研究者将完全采用路径依赖的动态规划方法来实现局部单调演化方程的最优控制。这一大类方程涵盖了二维纳维-斯托克斯方程和驯服的三维纳维-斯托克斯方程。此外，这项研究为无限维控制问题的最优综合这一广泛开放的问题提供了新的方法论。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查进行评估，被认为值得支持标准。